It's the Enterprise; that's what that whole scene with Jim Kirk stopping in the dark on his motorcycle was about. That's the moment he fell in love with the Enterprise. The literal light in his eyes is the reflection of the ship.
We don't see direct evidence that the ship in the dark would be the Enterprise. She's merely a ship of that design.
We do see that the ship later seen in daylight is the Enterprise. But we could in theory say that after the fateful night, Kirk rode his bike all the way to San Francisco (he's the type to do that sort of riding), looked around to see where Starfleet might be, spotted the local shipyards (which had something like six of these babies under construction, including the Enterprise), drove there, and boarded the shuttle for the ride to the campus.
Timo Saloniemi
In the establishing shot of the bar, we see the Enterprise being built in background. And that's where he drives to.